Exercise 2.1

A. Using the Famous Forms Method

Write the forms of the following arguments. Then use the Famous Forms Method to determine if the arguments are valid. If the validity of an argument cannot be determined in this way, say that it cannot be determined.

1. You ate it. If you ate it, you’d find out it was spicy. So you found out it was spicy.
2. If you flip the switch, you’ll see sparks. And if you see sparks, you’re doing it wrong. So if you flip the switch, you’re doing it wrong.
3. If God has a reason for such horrible suffering, God has let us know the reasons for it. But God hasn’t let us know of any reasons for such horrible suffering. So God doesn’t have a reason for such horrible suffering.
4. There’s either a good chance that I’ll get the job or a good chance that I don’t have any hope for a happy life. If there’s a good chance I’ll get the job, I’ve set myself up for failure. If I don’t have hope for a happy life, I will fail inevitably. So either I’ve set myself up for failure or I’ll fail inevitably.
5. Some pleasures are not beneficial. Some pleasures are not worth experiencing. Those pleasures that are not worth experiencing are not beneficial.
6. If something exists, it has a cause. And if something has a cause, God is the ultimate cause. So if something exists, God is the ultimate cause.
7. The best things in life are free. And this was free. So this is one of the best things.
8. I didn’t get the SAT score I wanted. Either I got the score I wanted or I’m not getting into Harvard. So I’m not getting into Harvard.
9. Some food storage bins are carcinogenic. Some of those bins contain BPA, and all BPA products are carcinogenic.
10. If you clean your room, you should expect to set aside hours of your time. And you know that if you set aside hours of your time, you’ll miss out on the party. So if you want to clean your room, you want to miss out on the party.

B. Using the Famous Forms Method (Advanced)

Follow the same instructions as for exercise A.

1. No one will be at the party, because you can get into the party only if you know someone who is already there. And if you can get into the party only if you know someone who is already there, no one will be there.
2. I’ve never lived in Texas, because if I lived there I’d either own a big truck or cowboy boots, and I own neither a big truck nor cowboy boots.
3. If no one understands me, you will, because no one else understands me. And if no one else understands me, you’ll know it and try. And if you know it and try, then you will understand me.
4. Either Taylor will play goalie and we’re sure to win, or Jordan will play goalie. Jordan won’t play because she is injured. So Taylor will play and we’re sure to win.
5. If the company’s advertising isn’t deceptive, then it did not deceive reasonable people or most people. The company’s advertising did deceive most people. So the company’s advertising is deceptive.
6. Jesus is either Lord or a liar or lunatic. That’s because either Jesus’s claims to be God are true or Jesus made false claims to be God. If he made false claims to be God, he is a liar or lunatic. If Jesus’s claims to be God are true, then Jesus is Lord.
7. Here, what follows the “and” is an independent cause or an item in a list. You’re supposed to put a comma before the “and” only if what follows the “and” is an independent clause or an item on a list, and you are supposed to put a comma before the “and.”
8. I am either complimenting you or saying something that isn’t a compliment. If I am complimenting you, you’ll construe what I say as an insult. If I’m saying something that isn’t a compliment, you’ll construe what I say as an insult. So either you’ll construe what I say as an insult, or you’ll construe what I say as an insult.
9. If you learn about time management, you’ll spend your time listening to a lecture and you’ll learn what you shouldn’t spend your time doing. If you spend your time listening to a lecture and learn what you shouldn’t spend your time doing, you’ll realize you should not have spent your time learning about time management. So if you learn about time management, you’ll learn you shouldn’t have spent your time learning about time management.
10. In heaven, we can choose to do morally bad things, or we can only do good things. If we can choose to do bad things in heaven, then it is likely that someone will do morally bad things in heaven. And if we can only do good things in heaven, we don’t have any freedom in heaven. So either it’s likely that someone will do morally bad things in heaven or we don’t have any freedom in heaven.

Exercise 2.2

A. Identifying Argument Forms

Identify the form of each of the following arguments that is most relevant for employing the Counterexample Method.

1. No tigers are striped things. Some crustaceans are striped things. So no crustaceans are tigers.
2. Some piano music is music from Beethoven. All music by Mozart is piano music. Therefore, some music by Mozart is music from Beethoven.
3. If you want it enough and take action to get it, you’ll be successful. You’ll fail. So you don’t want it enough.
4. If corporations have a responsibility, it is to make a profit. That’s because if corporations have a responsibility, their responsibility is just to do what their shareholders want them to do, and if corporations have a responsibility to do what their shareholders want them to, corporations have a responsibility to make a profit.
5. Some unfulfilled desires are painful things. Some unfulfilled desires are not bad things. So some painful things are not bad things.
6. Doctors should not honor patients’ wishes not to receive medical treatment. If doctors should honor patients’ wishes not to receive medical treatment, they would violate their obligation to treat patients, and doctors should not violate their other obligations.
7. All license holders are drivers. No one under sixteen is a license holder. So no one under sixteen is a driver.
8. Either you tell me your secrets or we won’t be friends. You tell me your secrets, so we’ll be friends.
9. Some worship is a good thing. Some good things are things that are pleasing to God. So all worship is a thing that is pleasing to God.
10. Apple Watch adoption in the United States will rise when Apple will advertise to consumers. Apple will advertise to consumers only when there are enough merchants who are willing to accept Apple Pay. So when there are enough merchants who are willing to accept Apple Pay, Apple Watch adoption in the United States will rise.

B. Counterexamples

Construct a good counterexample to each of the following argument forms.

1. If P or Q, then R or S. Not R. So neither P nor Q.
2. Some A are not B. Some B are not C. So some C are not A.
3. If P and Q, then if R then S. Not S, so not P or not Q.
4. All A are B. All A are C. So all B are C.
5. If P, then both R and Q. So either Q or P.
6. No A are B. No B are C. So some A are not C.
7. P or Q. If P, then R. If Q, then S. So R and S.
8. All A are B. Some B are C. So some A are C.
9. If P, then S. If S, then R. If R, then Q or T. Q. So P.
10. All A are B. Some B are not C. So some A are not C.

Exercise 2.3

A. Venn Diagrams

Use Venn diagrams to evaluate the following categorical syllogisms. If the Venn diagram allows you to conclude that the syllogism is valid, indicate that it is valid. If the diagram does not, indicate that it does not.

1. Some cases of anxiety are cases it is reasonable to be in. No cases that it is reasonable to be in are cases that involve irrational desires. So some cases of anxiety are not cases that involve irrational desires.
2. Some blood transfusions are not permissible procedures. All permissible procedures are procedures that do not violate the conscience of the patient. So some blood transfusions are procedures that violate the conscience of the patient.
3. No leggings are pants. No pants are capris. So no leggings are capris.
4. Some steelworkers unions are not bonded unions. All bonded unions are unions that are not liable to fraud. So some unions that are not liable to fraud are steelworkers unions.
5. Some computers are machines that are not protected from viruses. Some machines that are not protected from viruses are machines that are susceptible to hacking. So some computers are machines that are susceptible to hacking.
6. No watch charging ports are wireless ports. All wireless ports are ports that use magical magnetic powers. So no ports that use magical magnetic powers are watch charging ports.
7. All logic problems are an opportunity to practice. All opportunities to practice are a chance to make you a better person. So some chances to make you a better person are logic problems.
8. All foods with sugar are unhealthy foods. No unhealthy foods are foods worthy of your time. So no foods with sugar are foods worthy of your time.
9. All universities are places that overcharge tuition. Some places that overcharge tuition are places worth the investment. So some universities are places worth the investment.
10. Some football games are well-attended events. All well-attended events are events that need food. So some football games are events that need food.

B. Transforming Arguments

Transform each of the following arguments into a categorical syllogism. Then use a Venn diagram to evaluate it. If the diagram allows you to conclude that the argument is valid, indicate that the argument is valid. If it does not, indicate that it does not.

1. Some people have diseases. All things with diseases are sick. So some people are sick.
2. This report is fraudulent. All fraudulent things are produced by bad people. So this report is produced by a bad person.
3. No good people are Nazarites. Jesus is a Nazarite. So Jesus is not good.
4. All sales that are up this quarter have come from our software divisions. The software divisions receive only recurring revenue. So recurring revenue accounts for all our sales that are up.
5. Some transactions involving bribes are not impermissible. All shipments through Thailand are transactions involving bribes. So some shipments through Thailand are not impermissible.
6. No insider information can be used for stock trading purposes. All information that can be used for stock trading relies solely on publicly available information, and no insider information relies solely on publicly available information.
7. All people who eat contaminated meat get sick. No one who is sick can play basketball well. So if anyone eats contaminated meat, they won’t be able to play basketball well.
8. Some students cheat on tests. No ethics students cheat on tests. So there is at least one student who is not in ethics.
9. Some breakfast sandwiches contain avocado, but all sandwiches containing avocados are reserved for gourmet dishes. So some breakfast sandwiches are a gourmet dish.
10. At least one person lived a completely righteous life. Everyone who lives a completely righteous life has not sinned. So not everyone has sinned.

Exercise 2.4

A. Proofs with Rules of Implication

Construct proofs of the following symbolized arguments using the rules of implication.

1. Q * R. R -> ~Q. So Q

2. ~(P * Q) v ((R v T) -> Q). ~~(P * Q) * (R v T). So Q

3. (P v S) -> R. (P -> Q) -> R. (P v S) v (P -> Q). So R v R

4. P -> (P -> Q). P. So Q.

5. ~P -> ~ ~ Q. S -> ~Q. T -> ~P. T. So ~S

6. (A v T) -> B. R -> (C * ~R). R. R -> (A v T). So (B v (C * ~R)) * (A v T)

7. A v (B v C). A -> B. B -> R. C -> S. ~B. So R v S

8. (R * S) * (T * V). So S v T

9. S. S -> ~(T v L). L -> (T v L). So ~L

10. S -> T. T -> L. T -> P. ~P * S. So L

B. Proofs with Rules of Implication and Equivalence

Construct proofs of the following symbolized arguments using the rules of implication or the rules of equivalence.

1. ~(P -> Q). So P * ~Q

2. ~(~P -> (Q v (R -> Q))). So R * ~(P v Q)

3. ~(P * (~T -> R)) * R. So ~P

4. (~P v ~Q) v Q. So ~ ((P * Q) * ~Q)

5. ~P -> (Q v R). So ~(P v Q) -> R

6. (P * P) v (P * R). R <–> P. So P

7. ~(~P v Q) -> ~P. So P -> Q

8. (P v Q) v R. ~R * ~P. So Q

9. ~R * (~S -> (Q -> R)). So ~Q v S

10. Q. So (P v ~P) -> Q

C. Indirect Proofs

The following are either arguments or theorems. For the arguments, construct proofs using the rules of implication, equivalence, or indirect proof. For the theorems (the examples with only one line), give a proof of the theorem from no premises.

1. ~(P * ~P)

2. P v ~P

3. (P v Q) <-> ~(~P * ~Q)

4. (P -> Q) <-> (~P v Q)

5. (P v Q) v R. Q -> (R v P). So ~P -> R

6. P -> ((Q v R) -> P)

7. P -> (~Q v ~S). (~Q -> R) * (~S -> T). ~(R v T). So ~P

8. (P * ~Q) v R. P -> R. So R

9. R -> ((~S v ~T) * Q). So T -> (R -> Q)

10. R * (S -> T). So S -> ~(~R v ~T)

D. Proving English Arguments

Symbolize the following arguments for purposes of using the Proof Method. Then construct proofs of them using any of the rules covered in this section.

1. If price gouging is morally permissible, a company should be allowed to price excessively high a good that is necessary for peoples’ fundamental well-being. But I shouldn’t choose which of two drowning victims to save on the basis of who can pay the highest price. And if I shouldn’t chose which of two drowning victims to save on the basis of who can pay the highest price, a company should not be allowed to price a good necessary to people’s fundamental well-being excessively high. So price gouging is not morally permissible.
2. We are not responsible for any of our actions. Either all of our actions are determined by the past together with the laws of nature, or they are not. If they are determined by the past together with the laws of nature, our actions are determined before we were even born. If they aren’t determined by the past together with the laws of nature, they are the result of random quantum fluctuations, and if they’re either the result of random quantum fluctuations or determined before we were born, then we aren’t morally responsible for them.
3. Death is not bad for you, because if God exists, God is loving. And a loving God helps restore people back to spiritual health, not cause them everlasting torment. Plus, if God doesn’t exist, death is just the end of our existence, and if it’s the end of our existence, we’re not around to experience anything bad. And whether we don’t exist to experience anything bad or we exist and are restored to spiritual health, death isn’t bad for you.
4. Justice will not be served in this trial. Either the defendant will be declared guilty or he will not be. If he will be, he will receive fewer than ten years in prison, and if he will not be declared guilty, he will not receive any prison time. If he will receive fewer than ten years in prison or he will not receive any prison time, he will spend time out of prison. If he will spend time out of prison, justice will not be served in this trial.
5. Whether something is true does not depend on whether people agree that it’s true. This is because people do not agree that it is true that whether something is true depends on whether people agree to its truth, and if it’s true that both whether something is true depends on whether people agree that it’s true and people do not agree that it is true that whether something is true depends on whether people agree to its truth, then whether something is true does not depend on whether people agree that it’s true.
6. God exists, and here’s why. There are things in the universe that do not have to exist. If there are things that do not have to exist, something causes these things to exist. If something causes these things to exist, either they cause their own existence or something else causes their existence. They don’t cause their own existence. And if something else causes them to exist, either they are ultimately caused by something that has to exist or only by things that don’t have to exist. If they are only ultimately caused by things that don’t have to exist, there is an infinite number of causes. There isn’t an infinite number of causes. If things are ultimately caused by something that has to exist, then God exists.
7. If utilitarianism is true (note: utilitarianism is the view that an action is right if and only if it results in the maximum amount of pleasure for the most people), then: if a doctor could harvest the organs of one healthy patient without the patient’s consent to save five sick patients in need of organs and the doctor would not feel remorse nor would anyone else know about the operation, then the doctor does the right thing. But it is not the case that if a doctor could harvest the organs of one healthy patient without the patient’s consent to save five sick patients in need of organs and the doctor would not feel remorse nor would anyone else know about the operation, then the doctor does the right thing. So utilitarianism is false.
8. If fetuses have a right to life, then abortion is not morally permissible. And if fetuses do not have a right to life, neither do infants, as long as fetuses and infants have relatively similar mental capacities. Fetuses and infants do, in fact, have similar mental capacities. So if abortion is morally permissible, infants do not have the right to life.
9. If today is Tuesday or Thursday, then Jane will attend logic and Jane will attend ethics. So either today is neither Tuesday nor Thursday, or Jane will attend logic.
10. Capital punishment is impermissible. For capital punishment is permissible only if it is an effective deterrent for the offender. Yet capital punishment ends the life of an offender. And if capital punishment ends the life of an offender or makes committing the crime appealing for the offender, it is not an effective deterrent for the offender.

Exercise 2.5

A. New Vocabulary

Symbolize the following statements using the symbols of the Proof Method and the new symbols introduced for the Expanded Proof Method. Pick your own symbols for predicates, variables, and constants within the parameters set by this section.

1. Every freshwater fish also has scales.
2. You must be twenty-one years old to drink alcohol in the United States.
3. Some people believe a deity exists.
4. At least one person in this room has a birthday in September.
5. You should let me out of this room only if I either say the password or you hear scratching sounds.
6. If anyone loves, Bob does.
7. Bob is loved, but no one else is.
8. The treasure is here, and either it will be found or everyone will be poor.
9. Every good and perfect gift comes from God.
10. There isn’t a thing in the world that isn’t either thought of by God or desired by someone else.

B. Proofs for Symbolized Arguments

Provide a proof for each of the following symbolized arguments using the Expanded Proof Method.

1. ∀ x(Fx <-> Gx). So ∀x(Fx) <-> ∀x(Gx)

2. ∃x(Fx * Gx) -> (∃x(Fx) * ∃x(Gx))

3. ∀x(Fx * Gx) <-> (∀x(Fx) * ∀x(Gx))

4. ~∀x(Fx <-> Gx) v (∀x(Fx) <-> ∀x(Gx))

5. ∃x(Fx) v ∃x(Gx). ∀x(Fx -> Gx). So ∃x(Gx)

6. ∀x~(Px -> Qx). ∃x(Qx). So Ra

7. ∀x(~Px v (Qx v Tx)). ∀x~(~Sx v Tx). So ∀x(Px -> Qx)

8. ∀y(Fa -> (∃x((~Gx) -> Gy)). ∀x(Gx -> Hx). ∀x(~Jx -> ~Hx). So ∃x(~Jx) -> (~Fa v ∀x(Gx))

9. ∃x(Fx * Ga). ∀x(Fx -> Hx). So ∃x(Ga * (Fx * Hx))

10. ∀x(Gx -> ∃y(Fy * Hy)). So ∀x(~Fx) -> ~∃yGy

C. Proofs for English Arguments

Symbolize each of the following English arguments using the symbols of the Proof Method and the Expanded Proof Method. Then provide a proof.

1. God isn’t blameworthy for the Holocaust. That’s because although the Holocaust was an instance of suffering, and God allowed the Holocaust, there is a morally sufficient reason for every instance of suffering, and if an event has a morally sufficient reason for occurring, that event is not, on balance, bad. God is blameworthy for an event only if that event is, on balance, bad.
2. There isn’t anyone who has the worst evil happen to them. If someone is in hell, the worst evil happens to them. So no one is in hell.
3. All statements worth saying are true. Every false statement is untrue. So there is not even one untrue statement worth saying.
4. Everyone who has performed every great moral good has asked for forgiveness. But someone asks for forgiveness if and only if they have done something wrong. If there are any saints, they have performed every great moral good and have never done anything wrong. So there are no saints.
5. Bill will be at the party if either Melinda or Jordan will be at the party. Since Bill is a friend of mine, I’ll have a friend at the party if Jordan is there.
6. Every time someone reflects, they are able to evaluate all of their beliefs. Whenever someone is able to evaluate all of their beliefs, they’re able to evaluate whether they know. So either nobody reflects at all, or it’s false that when someone reflects they aren’t able to evaluate whether they know.
7. God exists and there is evil. So it is false that if there is evil, then there is no God.
8. Every available house on the market is either falling apart or over your price limit. You will buy a house, but you won’t buy anything over your price limit. So you’ll buy a house that is falling apart.
9. All business that is done overseas is vetted by foreign officials. Business that is vetted by foreign officials is not private. So Google’s business overseas is not private.
10. The president’s statement to the media was not a lie. A statement is a lie only if the statement is expected to be true by the audience. The audience to the president’s statement, however, did not expect the statement to be true.